Fitting splines to a parametric function

Preface......Page 7
Contents......Page 10
1 Introduction......Page 12
 2.1 Definition of Error Function......Page 14
 2.2 Character of Solution......Page 15
 2.3 Optimization of F(a, u)......Page 16
 References......Page 22
3 General Properties of Splines......Page 23
 References......Page 27
4 ODF Using a Cubic Bézier......Page 28
 4.1 Fitting a Function with a Double Inflection Point......Page 29
 4.2 Initializing a Cubic Bézier......Page 30
 4.3 Optimizing the Fit......Page 31
 4.5 Character of Coalescing Solutions......Page 33
 References......Page 35
 5.1 Center of Mass Fit......Page 36
 5.2 Example of Two Types of Merge/Crossover......Page 39
 5.3 Response to Change in g(t)......Page 41
 5.4 Distinguishing Between Type 1 and Type 2 Events......Page 43
 References......Page 45
6 ODF Using a 5-Point B-Spline......Page 46
 6.2 Basis Functions of a 5-Point B-Spline......Page 47
 6.3 Decomposition into Two Bézier Segments......Page 48
 6.4 ODF Results for a 5-Point B-Spline......Page 50
 7.1 Initializing a 6-Point B-Spline......Page 52
 7.2 Basis Functions of a 6-Point B-Spline......Page 53
 7.3 Decomposition into Three Bézier Segments......Page 54
 7.4 ODF Results for a 6-Point B-Spline......Page 55
 8.1 Initializing a Quartic Bézier......Page 58
 8.2 ODF Results for a Quartic Bézier......Page 59
 8.3 Enumeration of Solutions......Page 61
 8.4 Abnormal Truncation of Solutions......Page 63
 8.5 Topological Comparison of Béziers and B-Splines......Page 64
 References......Page 65
9 ODF Using a Beta2-Spline......Page 66
 9.1 Rendering a Beta2-Spline......Page 67
 9.2 Nonlinear Effects Due to ψ......Page 70
 9.4 ODF Results for a Beta2-Spline......Page 71
 References......Page 74
 10.1 Rendering a Beta1-Spline......Page 75
 10.2 Nonlinear Effects Due to  β1......Page 76
 10.3 Continuity of the Beta1-Spline Integrands......Page 77
 10.4 ODF Results for a Beta1-Spline......Page 79
 10.5 Numerical Problems with the Beta1-Spline......Page 81
11 Conclusions......Page 84